Complete Mathematics

417 Pages·2010·1.51 MB·English

by  Johnson, Trevor

Complete Mathematics

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Complete
Mathematics
Trevor Johnson and
Hugh Neill

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First published in UK 2001 by Hodder Education, part of
Hachette UK, 338 Euston Road, London NW1 3BH.

First published in US 2001 by The McGraw-Hill Companies, Inc.

This edition published 2010.

Previously published as Teach Yourself Mathematics.

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Hodder Headline.

Copyright © 2001, 2003, 2008, 2010 Hugh Neill and
Trevor Johnson

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Impression number
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Year

2014 2013 2012 2011 2010

v
Contents

Contents

Personal introduction
xi

Only got a minute?
xii

Only got fi ve minutes?
xiv

Only got ten minutes?
xvi
1
Number
1
1.1 Introduction – place value
1
1.2 Arithmetic – the four operations
3
1.3 Order of operations
4
1.4 Problems which use arithmetic
7
1.5 Special numbers
8
1.6 Multiples, factors and primes
10
2
Angles
15
2.1 Introduction
15
2.2 Angles
16
2.3 Measuring and drawing angles
18
2.4 Using angle facts
20
2.5 Parallel lines
22
2.6 Bearings
25
3
Fractions
29
3.1 Introduction
29
3.2 What is a fraction?
29
3.3 Which fraction is bigger?
31
3.4 Simplifying fractions
31
3.5 Improper fractions
33
3.6 Adding and subtracting fractions
34
3.7 Multiplication of fractions
36
3.8 Fractions of quantities
37
3.9 Division of fractions
38
3.10 A number as a fraction of another number
40
4
Two-dimensional shapes
44
4.1 Introduction
44
4.2 Triangles
44
4.3 Constructing triangles
48

vi

4.4 Quadrilaterals
50
4.5 Polygons
52
4.6 Interior and exterior angles
53
4.7 Symmetries of regular polygons
55
4.8 Congruent shapes
55
4.9 Tessellations
57
5
Decimals
63
5.1 Introduction
63
5.2 Place value
64
5.3 Converting decimals to fractions
66
5.4 Converting fractions to decimals 1
67
5.5 Addition and subtraction
68
5.6 Multiplication of decimals
69
5.7 Division of decimals
72
5.8 Converting fractions to decimals 2
74
6
Statistics 1
80
6.1 Introduction
80
6.2 Collection of data
81
6.3 Pictograms
82
6.4 Bar charts
83
6.5 Pie charts
87
6.6 Line graphs
89
6.7 Scatter graphs
93
6.8 Discrete and continuous data
96
6.9 Grouping data
97
7
Directed numbers
104
7.1 Introduction
104
7.2 Ordering directed numbers
105
7.3 Addition and subtraction
106
7.4 Multiplication and division
108
7.5 Using a calculator
111
8
Graphs 1
113
8.1 Coordinates
113
8.2 Straight-line graphs
116
8.3 Lines parallel to the axes
121
9
Measurement
125
9.1
The metric system
125
9.2
Imperial units
127

vii
Contents

9.3
Converting between metric and imperial units
129
9.4
Choosing suitable units
130
10 Perimeter and area
134
10.1
Perimeter
134
10.2
Area
136
10.3
Area of a rectangle
138
10.4
Area of a parallelogram
141
10.5
Area of a triangle
141
10.6
Area of a trapezium
144
11 Algebraic expressions
148
11.1
Introduction – what is algebra?
148
11.2
Writing expressions
148
11.3
Simplifying expressions
151
11.4
Evaluating expressions
153
11.5
Squaring
155
11.6
Brackets
155
11.7
Factorizing expressions
158
11.8
Indices
159
11.9
Laws of indices
160
11.10 Simplifying expressions with indices
163
12 Approximation
169
12.1
Introduction
169
12.2
Rounding whole numbers
169
12.3
Rounding with decimals
171
12.4
Signifi cant fi gures
174
12.5
Estimates
176
12.6
Rounding in practical problems
178
12.7
Accuracy of measurements
179
13 Equations 1
182
13.1
Introduction
182
13.2
Finding missing numbers
183
13.3
Solving linear equations
183
13.4
Equations with brackets
188
13.5
Solving problems using equations
189
13.6
Solving inequalities
191
14 Percentages
195
14.1
Introduction
195
14.2
Percentages, decimals, fractions
195

viii

14.3
Percentages of quantities
199
14.4
Increasing and decreasing quantities
201
14.5
One quantity as a percentage of another quantity
202
14.6
Percentage increases
203
14.7
Using multipliers
205
15 Formulae
209
15.1
What is a formula?
209
15.2
Evaluating terms other than the subject
211
15.3
Changing the subject of a formula
213
16 Circles
216
16.1
Introduction
216
16.2
Circumference of a circle
218
16.3
Area of a circle
221
16.4
Two properties of circles
224
17 Probability
229
17.1
Introduction
229
17.2
Relative frequency
230
17.3
Probability of a single event
231
17.4
Two events
234
17.5
Tree diagrams
239
17.6
Expected frequency
243
18 Three-dimensional shapes
246
18.1
Introduction
246
18.2
Nets and surface area
249
18.3
Volume of a cuboid
251
18.4
Volume of a prism
253
18.5
Weight of a prism
256
19 Ratio and proportion
260
19.1
What is a ratio?
260
19.2
Scales
262
19.3
Using ratio
265
19.4
Direct proportion
267
19.5
Inverse proportion
269
20 Pythagoras’ theorem and trigonometry
273
20.1
Pythagoras’ theorem
273
20.2
Using Pythagoras’ theorem
274
20.3
Proof of Pythagoras’ theorem
278
20.4
Pythagoras’ theorem problems
279

ix
Contents

20.5
Trigonometry
280
20.6
The tangent ratio
281
20.7
Values of the tangent
282
20.8
Using tangents
283
20.9
Sine and cosine
285
21 Indices and standard form
292
21.1
Indices
292
21.2
Laws of indices
293
21.3
Prime factors
294
21.4
Highest common factor
295
21.5
Lowest common multiple
297
21.6
Standard form – large numbers
298
21.7
Standard form – small numbers
301
21.8
Standard form calculations
302
22 Statistics 2
306
22.1
Averages
306
22.2
The mode
306
22.3
The median
307
22.4
The mean
309
22.5
The range
312
22.6
Frequency tables
313
22.7
Grouped frequency tables
315
23 Graphs 2
322
23.1
Equations of straight lines
322
23.2
Drawing straight-line graphs
325
23.3
The gradient of a straight line
328
23.4
Curved graphs
331
24 Equations 2
337
24.1
Simultaneous equations
337
24.2
Algebraic methods
339
24.3
Quadratic expressions
343
24.4
Factorizing quadratic expressions
345
24.5
Quadratic equations
348
24.6
Solution by factorizing
349
24.7
Cubic equations
352

Answers
358

Taking it further
388

Index
393

x

Credits

Front cover: © Digifoto Silver/Alamy

Back cover: © Jakub Semeniuk/iStockphoto.com, © Royalty-
Free/Corbis, © agencyby/iStockphoto.com, © Andy Cook/
iStockphoto.com, © Christopher Ewing/iStockphoto.com,
© zebicho – Fotolia.com, © Geoffrey Holman/iStockphoto.com,
© Photodisc/Getty Images, © James C. Pruitt/iStockphoto.com,
© Mohamed Saber – Fotolia.com

xi
Personal introduction

Welcome to Complete Mathematics!

Complete Mathematics aims to give you a broad mathematical

experience and a fi rm foundation for further study. The book will

also be a useful source of reference for homework or revision for

students who are studying a mathematics course.

This book is primarily aimed at students who do not have teacher

support. Consequently, the explanations are detailed and there

are numerous worked examples. Throughout this new edition are

author insight boxes which provide additional support, including

memory aids and tips on how to avoid common pitfalls. Also new

to this book are end of chapter summaries of all the key points and

formulae.

Access to a calculator has been assumed throughout the book.

Generally, a basic calculator is adequate but, for Chapters 20 and

21, a scientifi c calculator, that is, a calculator which includes the

trigonometric functions of sine, cosine and tangent, is essential. Very

little else, other than some knowledge of basic arithmetic, has been

assumed.

To derive maximum benefi t from Complete Mathematics, work

through and don’t just read through the book. This applies not only

to the exercises but also to the examples. Doing this will help your

mathematical confi dence grow.

The authors would like to thank the staff of Hodder Education

for the valuable advice they have given.

Trevor Johnson, Hugh Neill, 2010

xii

Only got a minute?

Mathematics is probably the most versatile subject there

is. It has applications in a wide range of other subjects

from science and geography through to art. In science,

without mathematics there would be no formulae or

equations describing the laws of physics which have

enabled the technology that we use each day to develop.

Mathematics enables psychologists to fi nd out more about

human behaviour, looking for correlations between sets

of statistics. Doctors and pharmaceutical companies use

mathematics to interpret the data resulting from a drug

trial. An understanding of mathematics is essential in

architecture and engineering. Mathematics is also used to

model situations such as the stock market or the fl ow of

traffi c on motorways.

As well as being a useful subject, mathematics

is also intrinsically beautiful. Many famous artists like

Kandinsky and Escher used mathematics as a basis for

their work – Kandinsky made extensive use of geometry

Only got a minute?
xiii

in his paintings and Escher is famed for his complicated

tessellations and optical illusions. Mathematics also

occurs naturally – in nature there are many examples

of mathematical patterns and symmetry. For example,

the patterns on the wings of a butterfl y or moth are

symmetrical, and the pattern of seeds on a sunfl ower

head, the scales of a pinecone and the spiral of a snail’s

shell all follow the same sequence of numbers – called

the Fibonacci sequence.

Maths really is ubiquitous and having an

understanding of the basics of mathematics is invaluable

to many – arguably all – aspects of modern life.

xiv

5 Only got fi ve minutes?

The word ‘mathematics’ originally comes from the Greek for
knowledge and science. To most people maths means arithmetic –
addition, subtraction, multiplication and division – which are the
building blocks for maths. Without these we couldn’t interpret large
sets of data, solve an equation or work out the area of a shape.

We live in a technological age and mathematics underpins all of
this technology – without mathematics none of the luxuries of
modern life that we all take for granted would be possible. It is
hard to think of an aspect of modern life that is not reliant in some
way on technology and therefore mathematics. Mathematics is
also intrinsic to many careers: civil engineers rely on mathematics
to work out whether the bridge they are building will actually
support a given load and surveyors use trigonometry to calculate
the distance between two points. Also, insurance companies use
complex probability tables for their life assurance policies, the
self-employed will need to use percentages to fi le their tax returns,
while pharmacists use formulae to work out the correct dosage for
a particular drug and sportspeople are constantly using forms of
intuitive mathematics.

Many people lack confi dence in mathematics and feel that they are
not very good in this subject. For some reason it seems acceptable
to say, ‘Oh, I’m hopeless at maths’, in a way that would not be
accepted about reading or writing. This lack of confi dence is often
rooted in negative experiences when at school – not ‘getting’ a
diffi cult concept, new idea or solution to a problem. However,
without realising it, every day we are using maths. Working out
which box of cereal is best value for money involves proportion.
Placing a bet on a horse requires an understanding of probability.
Calculating the best mobile phone tariff involves using simple
formulae. Working out how much tax you should pay for your